Shell Theory Research Articles

Development of axisymmetric shell model for creep deformation analysis of reactor pressure vessel lower head

Under severe accident conditions of nuclear power plants, the lower head of a reactor pressure vessel can be at risk of creep deformation and potential failure. This study presents a computational model that can represent the creep deformation of the lower head through the application of shell theory. To account for the large deformation due to creep, kinematic equations are derived and implemented. Particularly, this study provides details of the mathematical formulation, which were lacking in previous studies. The analysis results using the developed model are compared with those of finite element analysis, considering deformation history, stress distribution, and deformed shape. A dimensionless time for the creep deformation is proposed based on the results, which can characterize the effective strain at failure. Finally, the developed model is applied to OECD lower head failure test for validation.

Determining the effect of longitudinal stiffeners on the deformation of multilayer ellipsoidal shells under non-stationary loads

The object of this study is the deformation of multilayer ellipsoidal shells under non-stationary distributed load. It is proposed to reinforce the shell with longitudinal stiffeners to enhance the strength of it. The application of Timoshenko's theory of shells and cores enabled the investigation of the influence of longitudinal stiffeners on the deformation of multilayer ellipsoidal shells under non-stationary loads with the discrete positioning of the stiffeners. A mathematical model of shell oscillations under various types of short-term non-stationary loads was based on the Hamilton-Ostrogradsky variational principle. The numerical algorithm based on the application of the integral-interpolation approach to the construction of finite difference schemes in spatial coordinates, combined with an explicit finite-difference scheme for the time coordinate, was used to solve the task set. It has been determined that reinforcing stiffeners significantly affect the deformed state of the multilayer ellipsoidal shell. The "shell-stiffeners" relation was taken into account as a base for research. The maximum deformation ε11 of the smooth three-layer ellipsoidal shell was on average 1.4 times greater than the deformation ε11 of the reinforced three-layer ellipsoidal shell throughout the entire studied time interval. It was determined that over time the presence of reinforcing stiffeners has a more significant impact on the deformed state of the reinforced ellipsoidal shell. A distinctive feature of this research is the consideration of the discrete placement of reinforcing stiffeners which made it possible to investigate the impact of longitudinal stiffeners on the deformation of multilayer ellipsoidal shells under non-stationary loads. The results could be used for the investigation of applied problems in research organizations and design bureaus when designing shell structures

Shape optimization of non-matching isogeometric shells with moving intersections

While shape optimization using isogeometric shells exhibits appealing features by integrating design geometries and analysis models, challenges arise when addressing computer-aided design (CAD) geometries comprised of multiple non-uniform rational B-splines (NURBS) patches, which are common in practice. The intractability stems from surface intersections within these CAD models. In this paper, we develop an approach for shape optimization of non-matching isogeometric shells incorporating intersection movement. Separately parametrized NURBS surfaces are modeled using Kirchhoff–Love shell theory and coupled using a penalty-based formulation. The optimization scheme allows shell patches to move without preserving relative location with other members during the shape optimization. This flexibility is achieved through an implicit state function, and analytical sensitivities are derived for the relative movement of shell patches. The introduction of differentiable intersections expands the design space and overcomes challenges associated with large mesh distortion, particularly when optimal shapes involve significant movement of patch intersections in physical space. Throughout optimization iterations, all members within the shell structures maintain the NURBS geometry representation, enabling efficient integration of analysis and design models. The optimization approach leverages the multilevel design concept by selecting a refined model for accurate analysis from a coarse design model while maintaining the same geometry. We adopt several example problems to verify the effectiveness of the proposed scheme and demonstrate its applicability to the optimization of the internal stiffeners of an aircraft wing.

Natural Vibrations and Stability of Composite Cylindrical Shells Containing a Quiescent Fluid

This paper presents the results of investigation of the natural vibrations and stability of circular vertical multilayered cylindrical shells, fully or partially filled with a quiescent compressible fluid subjected to hydrostatic and external static loads. The behavior of the elastic structure and fluid medium is described based on the classical shell theory and Euler’s equations. The linearized equations of motion of the shell, together with the corresponding geometrical and physical relations are reduced to a system of ordinary differential equations with respect to new unknowns. The acoustic wave equation is transformed to a system of differential equations using the method of generalized differential quadrature. The solution of the formulated boundary value problem is developed using Godunov’s orthogonal sweep method. The dependences of the lowest vibration frequencies and critical external pressure on the ply angle and the filling level of two-layer and three-layer cylindrical shells are analyzed in detail. It is demonstrated that, in contrast to the ply angle and a given combination of boundary conditions, the lay-up scheme of composite materials plays different parts in the problems of maximizing the fundamental frequency of vibrations and extending the stability boundaries.

Analytical seismic model for a tunnel with segmental liner buried in the half-space soil

In this paper, by using the wave function expansion (WFE) and the expansion of the cylindrical wave into plane wave (ECPW) methods, an analytical method for the dynamic response of a circular tunnel with segmental liner buried in the half-space soil to harmonic elastic waves is developed. The liner of the tunnel is supposed to consist of several segments and joints. Both the segments and joints are treated as open cylindrical shells, and the segment and joint shells together thus form an equivalent continuous shell (ECS) liner, and the thin shell theory is utilized to describe its vibration. The scattered wave field due to the presence of the tunnel and boundary of the half-space soil is divided into two parts, namely, the direct scattered wave field and secondary scattered wave field. The expressions for the direct scattered cylindrical waves in the soil is determined via the WFE method, while the secondary scattered waves form the boundary of the half-space soil are obtained by the ECPW method together with the application of the boundary condition along the soil surface. Applying the cylindrical shell theory on the ECS liner and using the Fourier series expansions for the variables and parameters of the ECS liner along the azimuthal direction together with introducing the Fourier component constitutive relation for the ECS liner, a system of equations for the Fourier component ECS displacements are derived with the differential equations for the ECS liner displacements. By using the system of equations, the expressions for the free and scattered wave fields and the continuity conditions at the interface between the liner and soil, the system of equations for the ECS liner coupled with the soil are derived. By using the developed analytical method for the tunnel, some results of the ECS tunnel under different incident waves are given.

Convexity conditions for fiber-reinforced elastic shells

Convexity conditions of the Legendre–Hadamard and quasiconvexity type are derived in the context of a theory for fiber-reinforced shells based on Cosserat elasticity. These furnish two-dimensional shell-theoretic versions of their three-dimensional counterparts.

The Vibration Characteristics Analysis of Fiber-Reinforced Thin-Walled Conical–Cylindrical Composite Shells with Artificial Spring Technique

In this paper, the vibration characteristics (free vibration and forced vibration) of fiber-reinforced thin-walled conical–cylindrical composite shells (FTCCS) are analyzed by combining theory and experiment. Based on the classical shell theory and Kirchhoff–Love assumption, the overall structure of the FTCCS is theoretically modeled. The artificial spring technology is used to simulate the arbitrary boundary conditions at the joint, which is divided into a main spring and a secondary spring. At the same time, the displacement functions of the two sub-structure shells of the FTCCS are established by the orthogonal polynomial method, and the energy function of the FTCCS is constructed according to the continuity condition of the bolt joint. Then, the vibration characteristics of the FTCCS are solved using the Rayleigh–Ritz method and the modal superposition method. Finally, the TC300/epoxy resin-based fiber-reinforced thin-walled conical–cylindrical composite shells are selected as the research object, and the rationality of the mathematical model is verified by pulse, frequency sweeping and resonance excitation tests. The experimental results show that the errors between the natural frequency and the resonance displacement calculated by the theory and the experimental results are 1.75–4.76% and 3.79–9.28%, which confirms the rationality of the proposed model. By changing the length of the shell, the lamination angle and the semi-cone angle of the conical shell, the vibration characteristics parameters under different conditions are calculated to evaluate the influence of three physical parameters on the vibration characteristics of the FTCCS.

Analytical method for free vibration of steel pipe piles partially submerged in water and penetrated in layered soil

This paper presents a novel analytical method for dynamic characteristics of steel pipe piles submerged in water based on the thin shell theory by using the state space method. The state equation with respect to the length direction is first derived by idealizing the steel pipe pile in water as a thin cylindrical shell, where the soil around the pile is modelled as an elastic foundation with stiffnesses in three directions and the surrounding seawater is considered through added equivalent mass. The governing equations for the free vibration of steel pipe piles are derived after the expansion of Fourier series for the state vectors along the circumferential direction. The frequency equation for free vibration and its corresponding mode shapes are then obtained, which is used to analyze the dynamic characteristics of steel pipe piles in water. Due to the advantage of the state space method, arbitrary boundary conditions of the steel pipe pile are conveniently achieved as well as the effects of seawater, soil, the piling hammer and guiding frame. Finally, the present analytical method is verified and demonstrated in detail by several numerical examples, and results show its simplicity and efficiency.

Free vibration analysis of closed and open horizontal cylindrical shells utilizing coupled transfer function method and differential transform method

This article aims to analyze the free vibration of closed and open horizontal circular cylindrical shells based on the concept of state-space transformation in conjunction with the differential transform method (DTM). The governing equations of the cylinders were developed based upon Sanders’ thin shell theory. Then, the set of coupled governing differential equations transformed into nine first-order differential equations. Afterwards, the well-established semi-analytical scheme so-called DTM was employed to solve the differential equations. The numerical investigation of the present method was also comprehensively carried out by analyzing some closed and open cylindrical shells with different boundary and compatibility conditions. To ensure the correctness of the results, the outcomes were compared with experimental and other numerical results, especially with the well-known finite element method. Furthermore, some plots of two-dimensional and three-dimensional mode shapes for the first and second axial modes are depicted.

Theoretical modeling and analysis of an equipment–shell multiple-transmission-path system with attached liquid-filled pipe

Vibrations generated by mechanical equipment are the main source of radiation noise for ships. In actual systems, there are multiple transmission paths for mechanical vibrations from the equipment to shell. Common paths include vertical support structures, such as vibration isolators, and lateral attachment structures, such as pipes. However, most existing models only consider a single-path scenario. Multi-path systems have more complex modal and vibration transmission characteristics and introduce more complex coupling problems. There is a lack of theoretical modeling and physical analyses of common multiple transmission systems in ships. In this study, a theoretical model of an equipment–shell multiple-transmission-path system with an attached liquid-filled pipe is established as a typical multiple-transmission-path system for ships. In this model, the liquid-filled pipe is solved using a newly proposed traveling-wave method based on Kennard shell theory. Based on the theoretical model, the effects of each type of traveling wave on the pipe on the system modal characteristics, the mechanism behind each path on the system coupling characteristics, the influence of structural parameters on system modal and vibration transmission characteristics, and the proportion of vibration transmitted by the two transmission paths are analyzed. This study provides theoretical insights and guidance for practical vibration and noise reduction engineering.

Analysis of vibration characteristics of lattice-core sandwich annular spherical shells

This paper investigates the vibrational characteristics of lattice-core sandwich annular spherical shells. An effective analytical model, based on the Smeared Stiffener technique, is employed to integrate the stiffness contributions of the core with those of the shells. Helical stiffeners are modeled as beams capable of bearing axial forces and bending moments. The governing equations are derived from Donnell's classical thin shell theory. The Galerkin method is applied to extract the natural frequencies. To validate the analytical results and conduct a comprehensive parametric study, a 3D finite element model is developed using ABAQUS CAE software. Comparisons demonstrate a satisfactory agreement between the analytical and numerical results. Additionally, the effects of the spherical shell's geometric parameters, lamination angle, stiffener orientation angle, and various lattice core configurations are examined.

Mathematical modelling of stability problems for thin transversally graded cylindrical shells

The objects of consideration are thin linearly elastic Kirchhoff–Love-type open circular cylindrical shells having a functionally graded macrostructure and a tolerance-periodic microstructure in circumferential direction. The first aim of this contribution is to formulate and discuss a new mathematical averaged non-asymptotic model for the analysis of selected stability problems for such shells. As a tool of modelling we shall apply the tolerance averaging technique. The second aim is to derive and discuss a new mathematical averaged asymptotic model. This model will be formulated using the consistent asymptotic modelling procedure. The starting equations are the well-known governing equations of linear Kirchhoff–Love second-order theory of thin elastic cylindrical shells. For the functionally graded shells under consideration, the starting equations have highly oscillating, non-continuous and tolerance-periodic coefficients in circumferential direction, whereas equations of the proposed models have continuous and slowly-varying coefficients. Moreover, some of coefficients of the tolerance model equations depend on a microstructure size. It will be shown that in the framework of the tolerance model not only the fundamental cell-independent, but also the new additional cell-dependent critical forces can be derived and analysed.

Postbuckling behaviour of anisotropic shear deformable laminated cylindrical shells with general imperfection distribution subjected to torsion

Composite cylindrical shells belong to typical basic structure in engineering, whose buckling and postbuckling behaviours determine the stability and safety of overall structures under complex service loadings. In this paper, a nonlinear mechanical model for anisotropic laminated composite cylindrical shells with general distributed initial imperfection is derived to analyse the buckling and postbuckling behaviours under torsional loads, in which the general distributed imperfections are represented by a generic two-dimensional imperfection function reconstructed by using measured data. Based on a higher order shear deformation shell theory, the governing equations are established with consideration of von Kármán–Donnell-type of kinematic nonlinearity and the coupling effects of extension/twist, extension/flexural and flexural/twist. By using a two-step perturbation technique, a new set of explicit solutions is obtained to determine the buckling loads and postbuckling equilibrium paths of the torsional composite cylindrical shells. Furthermore, the internal quantitative relationship between the shear stress with torsional characteristics along with an associate compressive stress of anisotropic laminated shell is for the first time obtained. Numerical analyses are performed to validate the accuracy and effectiveness of the present explicit solutions, and obtain the postbuckling response of different types of imperfections, anisotropic laminated cylindrical shells with different values of shell parameters.

Study on fracture of hyperelastic Kirchhoff-Love plates and shells by phase field method

Thin walled structures such as plates and shells are widely used in many engineering fields. To Predict its fracture behavior is of great significance for integrity design and strength evaluation of engineering structures. Numerical simulation of the fracture behavior of hyperelastic plates and shells is a challenge due to complex kinematic description, hyperelastic constitutive relationship, geometric nonlinearity and the degradation on elastic parameter caused by fracture damage. Combining Kirchhoff Love (K-L) shell theory with the fracture phase field method, and numerically discretizing the first and second order partial derivatives of displacement field and phase field by using T-splines and meeting the requirements of K-L plate and shell theory for the C1 continuity of the shape function, a model for the isogeometric analysis numerical formulation of the phase field fracture in hyperelastic K-L plates and shells is established. The fracture failure behavior of hyperelastic K-L plates and shells under the uniform load and displacement load is simulated, and the effect of the Gaussian curvature on the fracture behavior of hyperelastic K-L shells is studied. The simulation results show that the present numerical scheme can effectively capture the complex crack propagation path of plates and shells under the uniform load, and the displacement field can effectively reflect the crack distribution of materials. The thin shell with negative Gaussian curvature shows the excellent fracture performance under the internal pressure, and can withstand the greater internal pressure.

A shell theory approach for the analysis of metal-FRP hybrid toroidal pressure vessels

This study focuses on the analysis of metal-FRP hybrid toroidal pressure vessels (TPV) using shell theory. The analytical approaches for the design of metal-FRP TPVs considering bending effects are currently not available. Invariably the designs are done based on the most simplified approach like linear membrane theory. In this work, a potential energy functional for the metal-FRP TPV considering the membrane and bending deformations is developed using Love's shell theory. The classical laminate theory is employed to determine the stresses/strains throughout the thickness of the FRP laminate. The Rayleigh-Ritz method is adopted to solve the proposed potential energy functional. A finite element analysis (FEA) is conducted using ABAQUS to validate the proposed analytical solution. In addition, a parametric analysis is carried out to understand the influence of various parameters viz. thickness of base metal, thickness of FRP, and ratio between radii of toroid and cross-section on bending deformations in the hybrid TPV. The results from the proposed solution are in good agreement with that from FEA. Therefore, the proposed solution can be used in the analysis and design of the metal-FRP TPV irrespective of any variations in the geometric parameters. It is found that the inclusion of bending deformations can improve the accuracy of solution and the bending deformations in the base metal decreases or become negligibly small with the increase in R/r ratio and thickness of FRP. The orientation of fibers can change the direction of failure in the base metal. The important design parameters that influence the yield pressure are thickness of base metal and FRP, orientation of fibers and R/r ratio.

Nonlinear resonant analyses of graphene oxide powder reinforced hyperelastic cylindrical shells containing flowing-fluid

The resonant responses are investigated for the graphene oxide powder reinforced hyperelastic cylindrical (GOPRHC) shells containing flowing-fluid, and the shells are subjected to the radial harmonic excitations. Employing the improved Donnell's nonlinear shell theory, Halpin-Tsai model, hyperelastic constitution relation, velocity potential theory and Lagrange equation, the governing equation of motions are obtained for the GOPRHC shells containing flowing fluid. The amplitude frequency and force amplitude curves are presented by using the harmonic balance method and the pseudo-arc length continuation method. The effects of three distributions of the GOP, weight fraction of the GOP and fluid velocity on the natural frequency for the GOPRHC shells are discussed. The influences of different parameters on the dynamical responses for the GOPRHC shells containing flowing-fluid are conducted, including the external excitation, weight fraction of the GOP, fluid velocity and structural parameters. The results show that the GOPRHC shells with Hypere-X distribution containing flowing-fluid have the largest frequency. The super-harmonic resonance responses appear and present the synchronization effects with the primary resonant responses in the GOPRHC shells containing flowing-fluid. The increases of external excitations, fluid velocity, weight fraction of the GOP and structural parameters enrich the resonant responses for the GOPRHC shells. Compared to the fluid-filled hyperelastic cylindrical shells, the existence of the flowing-fluid makes the GOPRHC shells more prone to the chaotic vibrations. The hysteresis phenomena of chaotic vibrations occur in the GOPRHC shells containing flowing-fluid.

Dynamic Analysis of 2D FG Porous Thick Cylindrical Shells Under Thermal Shock

In this research, a thick hollow cylindrical shell made of bidirectional functionally graded open cell porous materials under internal thermal shock according to the classical theory of linear thermo-elasticity is examined for the first time. The cylinder is made of a porous cellular material and its porosity varies along both radial and axial directions continuously. The governing motion equations are obtained by using 2D-axisymmetric theory of linear thermo-elasticity rather than shell theories. This theory represents thickness stretching and gives more precise results. Graded finite element method is employed to model the problem. Applying this method rather than conventional FEM leads to more accurate results, especially for dynamic analyses. The cubic higher order element is used for dividing the solution domain. To obtain transient temperatures, Crank–Nicolson algorithm is used and then the Newmark procedure is used to derive time responses of displacement and stress components. The time history of displacements and stress components for different radial and axial power law exponents, porosity coefficient, boundary conditions, length-to-thickness ratio and two different porosity patterns are investigated in detail. The obtained results show that thermal-induced vibration is generally caused by hoop stress, and frequency and amplitude of vibrations and velocity of stress waves are considerably influenced by the porosity distribution, porosity coefficient and power law exponents in both directions.

On the nonlinear buckling and postbuckling responses of sandwich FG‐GRC toroidal shell segments with corrugated core under axial tension and compression in the thermal environment

AbstractThis paper presents an analytical approach for buckling and postbuckling analysis for functionally graded graphene‐reinforced composite (FG‐GRC) toroidal shell segments with the trapezoidal or round corrugated core under the axial tension and compression. The considered shells are placed in a thermal environment and surrounded by an elastic foundation. Based on the von Kármán‐Donnell shell theory with geometrical nonlinearities, Stein and McElman approximation, and a homogenization technique for corrugated shells, the basic equations of shells are established. The previous homogenization technique is improved by adding the thermal forces in the internal force expressions. The shell‐foundation interaction is expressed using the model of the Pasternak assumption. The Ritz energy method is used to obtain the pre‐buckling and postbuckling behaviors of the shells, from which the critical buckling tensions and compressions can be investigated. The influences of FG‐GRC face sheets, corrugated core, and foundation on the buckling behavior of sandwich shells can be shown in the numerical analysis.Highlights Postbuckling of toroidal shell segments with the corrugated core is analyzed. A homogenization technique is improved for the thermal forces in the core. The shells are made of functionally graded graphene‐reinforced composite. The nonlinear Kármán‐Donnell shell theory and Ritz energy method are applied. Special effects of input parameters on the buckling behavior are investigated.

Exact solutions for clamped spherical and cylindrical panels via a unified formulation and boundary discontinuous method

Numerous works in both recent and historical literature have concentrated on formulating theories to perform static analysis on simply-supported shell structures. However, it is worth noting that obtaining analytical solutions for clamped boundary conditions presents a strong challenge. In this paper, closed-form solutions for clamped cross-ply laminated and sandwich shells are achieved by employing a robust and hybrid methodology not previously reported in the literature. The high versatility of the Carrera Unified Formulation (CUF), based on the Equivalent-Single-Layer (ESL) description, is utilized to implement several refined shell theories. The Principle of Virtual Displacements (PVD) is utilized to derive the strong form of the governing equations in terms of displacement variables. As the main novelty, these equations are solved by the Boundary Discontinuous Fourier-based method (BDM) which provides highly accurate analytical solutions. The validity and robustness of the proposed methodology are assessed through a detailed comparison with references available in the open literature, as well as with FEM 3D results obtained with commercial software. Furthermore, the stress recovery technique is exploited to fulfill zero-stress and interlaminar continuity (IC) conditions. The findings might be useful in training artificial intelligence (AI) models, which, for instance, could facilitate the development of digital twin structures.

Structural Study of Four-Layered Cylindrical Shell Comprising Ring Support

In this work, the vibration analysis of a layered, cylinder-shaped shell is undertaken. The structure of the shell layers is composed of functionally graded and isotropic materials. The vibrations of four-layered cylindrical shells with a ring support along the axial direction are investigated in this research. The two internal layers are composed of isotropic materials, and the external two layers are composed of functionally graded materials. The outer functionally graded material layers considered are stainless steel, zirconia, and nickel. The inner two isotropic layers considered are aluminum and stainless steel. The shell frequency equation is acquired by employing the Rayleigh–Ritz method under the shell theory of Sanders. The trigonometric volume fraction law is used to sort the functionally graded material composition of the FGM layers. The natural frequencies are attained under two boundary conditions, namely simply supported–simply supported and clamped–clamped.